EE 531 Mechatronics
EE 531 – Mechatronics Laboratory
Experiment # 3
Permanent-Magnet Direct-Current Machines
(Motors, Generators, and Motor-Generator System)
Purpose The objective of this laboratory is to study permanent-magnet motors and generators used in high-performance mechatronic systems. The results obtained from experiments and numerical analysis should be validated. In particular, permanent-magnet motors and generators will be comprehensively examined.
Specifications and Data The permanent-magnet motors and generators are high-performance permanent-magnet electric machines, and the catalog data is provided in the appendix for two different motors/generators (you need the following parameters for the machines =xxx ohm, = xxx H, =xxx V-sec/rad, =xxx N-m/A, J=xxx kg-m2 and =xxx N-m-sec/rad – you must measure the armature resistance for two electric machines and used the measured ra1 and ra2 in your calculations). The angular velocity is measured using the voltage generated by the taxogenerators, and, therefore, the voltage-velocity constant (so-called taxogenerator “sensitivity”) of the taxogenerator should be used (see the appendix). All data must be reported in the SI units.
Motivation and Significance
Permanent-magnet motors have been widely used in high-performance electric drives and servo-systems, while permanent-magnet generators are used in the power systems. Figure 1 illustrates the permanent-magnet DC machine which can be used as motors and generators.
Figure 1. Schematic representation of permanent-magnet DC machines
Mathematical Model Developments
To the equations of motion, a schematic diagram of permanent-magnet DC machines is illustrated in Figure 2.
Figure 2. Schematic diagram of permanent-magnet electric machines
(current direction corresponds to the motor operation)
Using Kirchhoff’s voltage law and Newton’s second law of motion, the differential equations for permanent-magnet DC machines can be easily derived. Assuming that the susceptibility is constant (in reality, Curie’s constant varies as a function of temperature), one supposes that the flux, established by the permanent magnet poles, is constant. Then, denoting the back emf and torque constants as , we have the following differential equations describing the transient behavior of the armature winding and tortional-mechanical dynamics
,
. (1)
An s-domain block diagram of permanent-magnet DC motors is illustrated in Figure 3.
Figure 3. Block diagram of permanent-magnet DC motors
The angular velocity can be reversed if the polarity of the applied voltage is changed (the direction of the field flux cannot be changed). The steady-state torque-speed characteristic curves obey the following equation
, (2)
and the torque-speed characteristics are illustrated in Figure 4.
Figure 4. Torque-speed characteristic curves for permanent-magnet motors
The steady-state torque-speed characteristic curves are available from catalogs and must be calculated using the data reported in the “Specifications” section.
Part I: Permanent-Magnet DC Motor
Assignment 1: Calculate and plot the torque-speed characteristics
1.1. Problem to be solved and laboratory procedure: Calculate and plot the torque-speed characteristic curves for a permanent-magnet DC motor studied. Plot the load characteristic for different applied voltages (using different values for ua, e.g., 20%, 40%, 60%, 80% and 100% of the rated ua) and different load values. Simply use the values for three different load values, which will be found in the next section. Plot these on the same graph.
1.2. Problem to be solved and laboratory procedure Plot the theoretical torque-speed characteristics using the motor Specifications in the MATLAB environment. Compare them with the experimental torque-speed characteristics, see Example 1.1.
Example 1.1 A family of the torque-speed characteristic curves are found applying (2). In particular, plugging the different values of the armature voltage in ,
the steady-state characteristics are found and plotted in Figure 5. The load characteristic, as given by , is documented in Figure 5. The MATLAB m-file developed to calculate and plot the steady-state operating characteristics is given.
MATLAB script (ch_5_02.m)
% parameters of a permanent-magnet DC motor
ra=2.7; ka=0.105;
Te=0:0.01:1;
for ua=5:5:30;
wr=ua/ka-(ra/ka^2)*Te;
wrl=0:1:100; Tl=0.05+0.0001*wrl.^2;
plot(Te,wr,'-',Tl,wrl,'-');hold on;
axis([0, 1, 0, 300]);
end; disp('End')
Figure 5. Torque-speed and load characteristics
Assignment # 2. Simulate permanent-magnet DC motors
Problem to be solved and laboratory procedure: Simulate a permanent-magnet DC motor studied. Plot the transient responses (motor behavior) for different applied voltages (using different values for ua) and load torques (Te= 0, 20, 40, 60, 80 and 100%).
Let us simulate permanent-magnet DC motors in SIMULINK (you can use “classical” MATLAB and ode solvers). Two linear differential equations must be used to model the motor dynamics (see Equations 1). In particular, we have
, .
Using an s-domain block diagram documented in Figure 3, a SIMULINK diagram is built and illustrated in Figure 6.
Figure 6. SIMULINK diagram to model permanent-magnet DC motors
Example 2.1
The motor parameters (coefficient of differential equations) must be assigned to perform numerical simulations. Let, the motor parameters are assigned to be: =5 ohm, = 0.01 H, =0.2 V-sec/rad, =0.2 N-m/A, J=0.0005 kg-m2 and =0.00001 N-m-sec/rad. Let the applied armature voltage is =50rect(t) V, and the load torque is =0.2rect(2t) N-m.
One downloads these motor parameters in the Command window by typing
»ra=5; La=0.01; ka=0.2; J=0.0005; Bm=0.00001;
The transient responses for the state variables (armature current and angular velocity ), are illustrated in Figures 7. It should be emphasized that the plot command was used. In particular, to plot the motor dynamics one uses
»plot(x1(:,1),x1(:,2)); xlabel('Time (seconds)'); title('Armature current ia, [A]');
»plot(x2(:,1),x2(:,2)); xlabel('Time (seconds)'); title('Velocity wr, [rad/sec]');
Figure 7. Permanent-magnet motor dynamics
Example # 2.2: Simulation of permanent-magnet DC motors in the state-space (optional)
Our goal is to simulate permanent-magnet DC motors in SIMULINK using the state-space form. The differential equations to model permanent-magnet DC motors were found to be
, .
That is, in matrix form we have .
Denoting the state and control variables to be , one finds
with initial conditions .
The output equations is .
Hence, , .
Using the State-Space block, the modeling can be performed. To attain the flexibility, the symbolic notations are used. The State-Space block is illustrated in Figure 8.
Figure 8. State-Space Block with parameters of permanent-magnet DC motors
The developed SIMULINK diagram is documented in Figure 9.
Figure 9. SIMULINK diagram to simulate the motor dynamics
The simulation can be performed assigning the motor parameters and initial conditions.
The following motor parameters are used: =5 ohm, = 0.01 H, =0.2 V-sec/rad, =0.2 N-m/A, J=0.0005 kg-m2 and =0.00001 N-m-sec/rad. The voltage is =50rect(t) V.
The initial conditions are: .
To perform the simulations, in the Command window we download the motor parameters and initial conditions as
»ra=5; La=0.01; ka=0.2; J=0.0005; Bm=0.00001; x10=5; x20=100;
Running simulation and using the following plotting statement
»plot(x2(:,1),x2(:,2)); xlabel('Time (seconds)'); title('Angular velocity wr, [rad/sec]')
the dynamics of the motor angular velocity results, see Figures 10.
Figure 10. Motor angular velocity waveforms, =50rect(t) V
Part II: Modeling and Analysis of Permanent-Magnet Direct-Current Motors and Generators
We study the permanent-magnet DC generators driven by the prime mover. Our goal is to thoroughly study the transient dynamics and perform the steady-state analysis. To solve these problems, a mathematical model should be derived, and simulations must be performed augmenting the generator and prime mover dynamics. Let us assume that the permanent-magnet DC motor is used as a prime mover, and the power generation system considered is illustrated in Figure 11.
Figure 11. Permanent-magnet DC generator driven by DC motor
The mathematical model for permanent-magnet DC generator circuitry dynamics, if the resistive load is studied, is found using Kirchhoff’s law
,
and Newton’s second law of motion maps the torsional-mechanical dynamics of the generator – prime mover system
,
where is the generator armature current; is the angular velocity of the prime mover and generator; and are the armature resistance and inductance of the generator rotor winding; is the load resistance; is the back emf (torque) constant of the generator; and are the viscous friction coefficients; and are the moments of inertia of the prime mover and generator.
The electromagnetic torque developed by the permanent-magnet DC motor (prime mover), which is denoted as , is , where is the armature current in the prime mover rotor winding; is the back emf (torque) constant of the prime mover.
The load torque for the prime mover is the generator electromagnetic torque, and we have
.
Thus, one obtains
.
The mathematical model for the prime mover circuitry dynamics is
,
where is the voltage applied to the armature winding of the prime mover; and are the armature resistance and inductance of the prime mover rotor winding.
We have
,
,
(3)
In the MATLAB environment, the mathematical model derived for the permanent-magnet DC generator - DC motor generation system is used to perform simulations. In particular, a SIMULINK diagram is developed based upon a set of differential equations (3). The resulting diagram is illustrated in the book, see Figure 5.33 (file name is pm_pmg_1.mdl).
Assignment # 3. Simulation and Analysis of the Motor-Generator System
(Section 5.8)
3.1. Problem to be solved and laboratory procedure: Simulate a permanent-magnet DC motor – generator system studied in the MATLAB environment. Plot the transient responses (motor and generator behaviors) for different applied voltages (using different values for uapm, e.g., 20, 40, 60, 80 and 100% of uapm max) and loads (load motor using the generator using different values for RL, e.g., RL= 12, 18, 24, 30 and ¥ ohm). Use Table 1 to enter the experimental data and calculations for different uapm and RL. Compare the experimental and theoretical (steady-state) results.
Using the motor-generator system studied, examine the transient responses and steady-state characteristics for different loads and different angular velocities of the prime mover, which depend upon the armature voltage applied uapm. Measure the induced voltage by the generator uL and iag for different loads and angular velocity. Report the steady-state data in Table 2 (the simulation results must be reported as the Figures).
Table 1 – DC Motor Measurements
(% of uapm max) / ua / RL / ia / ug / ig / (RPM) / (rad/s)
(calculated) / Te
(calculated) / Efficiency
(calculated)
20 / 12
40 / 12
60 / 12
80 / 12
100 / 12
20 / 18
40 / 18
60 / 18
80 / 18
100 / 18
20 / 24
40 / 24
60 / 24
80 / 24
100 / 24
20 / 30
40 / 30
60 / 30
80 / 30
100 / 30
20 / ¥ (no load)
40 / ¥ (no load)
60 / ¥ (no load)
80 / ¥ (no load)
100 / ¥ (no load)
TA Signature ______Date ______